Introduction to Particle-in-Cell Methods in Plasma Simulations

FSC

Introduction to Particle-in-Cell Methods in Plasma Simulations Chuang Ren University of Rochester The 2011 HEDP Summer School

1

Simulations are an important tool in scientific research FSC

•  Computer simulations are carried out to

Experiment

–  Understand the consequences of fundamental physical laws –  Help interpret experiments

•  Can provide detailed information which is difficult to measure

–  Design and predict new experiments

Simulation

Theory

2

PIC simulations are indispensible in HEDP research FSC

•  Outline –  What is PIC method? –  Numerical algorithm for 2-1/2D EM code –  An example: two-plasmon-decay instabilities

3

There are various ways to describe a plasma FSC

• 

The complete description is to specify x(t) & v(t) of each particle –  Klimontovich formalism

• 

Neglecting particle-correlation (collisions), we get the Vlasov equation –  Closed Vlasov-Maxwell eqs.

• 

 ∂ q v×B + v ⋅ ∇ + α (E + ) ⋅ ∇ v ] f α ( x, v ,t) = 0 ∂t mα c  ∇ ⋅ E = 4 π ∑ qα ∫ f α dv

[

α

∇×B=

The fluid description describes macroscopic

quantities –  Closure problem

(For details see textbooks, e.g. Krall & Trivelpiece)

1 ∂E 4 π + ∑ qα c ∂t c α

∇×E =−

€

∫ vf

α

 dv

1 ∂B c ∂t

  ∂nα ( x,t) + ∇ ⋅ (nαVα ( x,t)) = 0 ∂t  ∂ V ×B nα mα Vα + nα mαVα ⋅ ∇Vα − nα qα (E + α )+∇⋅P ∂t c = −∑ nα mα (Vα − Vβ ) < ν αβ > β

4

€

Particle simulations take the Klimontovich approach FSC Maxwell’s Eqs •  Particle methods simulate a plasma system by following a number of particle trajectories •  Essential physics can be captured with a much smaller number of particles than that in a real plasma

€

•  The charge/mass ratio and € charge density remain the same •  103~1011 particles used

•  Statistics of the model system is different from a real plasma •  The detailed description is computation-intensive

  x i,v i

  E( x i ),B( x i ) Newton’s Eq + Lorentz force

€

4πne ω = m

2

2 p

Invariant e/m and ne -> invariant ωp 5

€

The phase-space is efficiently represented in particle methods FSC

•  It is difficult to sample a 6D-

P

unoccupied phase space (wasted cells)

phase space by cells o  Difficult to solve Vlasov equations in high dimensions

o  PIC codes use particles to efficiently sample the phase space distribution

plasma distribution function

X 6

Calculating inter-particle forces directly is expensive FSC

•  number of calculations~ 100 Np2

108 particles, 100 Tflop/s  1 time step ~ 3 hrs  10000 time step sim. ~ 3 yrs

7

Particle-in-Cell Algorithms FSC

• 

Find (ρij,Jij) from (xk,vk)

• 

Solving Maxwell’s eqs on grid to obtain (Eij,Bij)

• 

Find Fk from (Eij,Bij)

• 

Number of calculatons for 1 step – 

Deposit charge/current ∝Np

– 

Advance fields ∝Ncells

– 

Interpolate forces and advance particles ∝Np

– 

Total ~ αNp + β(Ncells)

108 particles, 100 Tflop/s 64 x 64 x 64 cells  1 time step ~ 0.3 ms  10000 time step sim. ~ 3 s 8

The Simple & Straightforward PIC Scheme FSC

dp v ⎛ ⎞ = q ⎜ E + × B⎟ ⎝ ⎠ dt c

Integration of equations of motion, moving particles

Fk  uk  xk Weighting

( E , B )ij  Fk

Δt

Integration of Field Equations on the grid

( E , B )ij  Jij

Weighting

(x,u)k  Jij

∂E = 4π j − c∇ × B ∂t ∂B = −c∇ × E ∂t 9

Finite-size particles reduce collision FSC

•  Close-range collisions are much reduced by using the finite-size particles •  This compensates for the smaller number of particles used –  ν/ωp~(nλD3)-1

•  We are interested in nλD3>>1 regime •  PIC models are not collisionless –  Can be used to test collision operators

10

FSC

•  References on PIC methods –  J. M. Dawson, ‘Particle simulation of plasmas,’ Rev. Mod. Phys. 55, 403-447 (1983) –  R. W. Hockney and J. W. Eastwood, Computer Simulation Using Particles, McGraw-Hill, New York (1981) –  C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Simulation, Institute of Physics Publishing, Bristol, UK (1991)

11

FSC

Numerical Algorithms for a 2-1/2D Electromagnetic PIC Code

no spatial variation in z (half-dimension)

12

Advancing EM Fields FSC

•  The EM fields are advanced by Maxwell’s Equations –  The other 2 Maxwell’s eqs are satisfied as initial conditions

€ •  In 2D, the field components can be decomposed into 2 independent modes –  TE mode (k·E=0): Ez, Bx, By. –  TM mode (k·B=0): Bz, Ex, Ey.

∂t B = −∇ × E, ∂t E = ∇ × B − J. ∇ ⋅ E = ρ, ∇ ⋅ B = 0.

€

∂t Bx = −∂y E z ,∂t By = ∂ x E z ,∂ t E z = ∂x By − ∂ y Bx − J z . ∂t Bz = −(∂ x E y − ∂y E x ),∂ t E x = ∂y Bz − J x ,∂t E y = −∂x Bz − J y . 13

€

FSC

Centered difference can be used in both time & space domain

For TM mode 2 n−1/ 2 Bzni,+1/ − B z j i, j

Δt

=−

E xni,+1j +1/ 2 − E xni , j +1/ 2 Δt E yni+1 − E xni +1/ 2, j +1/ 2, j Δt

=

E yni +1/ 2, j − E yni −1/ 2, j Δx 2 2 Bzni,+1/ − Bzni,+1/ j +1 j

=−

Δy

Δy

(Ex,,Jx)

− j x i , j +1/ 2

2 Bzni +1,+1/j 2 − Bzni,+1/ j

Δx

+

E xni, j +1/ 2 − E xni , j −1/ 2

j

− j y i +1/ 2, j

Bz (Ey,, Jy)

i

TE mode is similar €

Bx

j

By (Ez, Jz)

i

14

Combining the Two Sets of Fields FSC

•  Two equal ways of putting both modes on one grid

j+1

Yee lattice j+1

or

(Ez,, Bz, Jz)

j+1/2 (E , B , J ) (E , B , J ) j i i+1/2 i+1 y,

y

y

x,

x

(Ez,, Jz)

j+1/2 (E , B , J ) (E , B , J ) j B i i+1/2 i+1 x,

x

y

x

y,

x

y

z

•  Leapfrog in time domain

E

x n-1

B,J

n

n-1/2

n+1/2

V

15

Time step is limited by the Courant condition FSC

• 

Plane EM Waves in Vacuum

• 

Dispersion relation from difference scheme

–  (E,B)=(E0,B0)exp(ik·x-iωt)

€

• 

Small ∆ t is required for stability in explicit schemes

• 

Large ∆ t can be achieved with implicit schemes –  Still need to resolve relevant physics

sin(ωΔt /2) ωΔt /2 sin(kx Δx /2) κ x = kx k x Δx /2 Ω=ω

ΩB = κ × E ΩE = −κ × B

2 ⎛ ωΔt ⎞ 2 ⎛ k x Δx ⎞ 2 ⎛ k y Δy ⎞ ⎜ sin 2 € ⎟ ⎜ sin 2 ⎟ ⎜ sin 2 ⎟ 2 2 2 Ω = c κ ⇒ ⎜ ⎟ ⎟ = ⎜ ⎟ + ⎜ ⎜ cΔt ⎟ ⎜ Δx ⎟ ⎜ Δy ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

€

1 1 1 > 2+ 2 2 c Δt Δx Δy 2

Courant Condition for Numerical Stability

€

16

The numerical dispersion relation contains modes with VphVph, unwanted Cerenkov emission can occur.

2

Numerical dispersion (c∆t/∆x=0.6)

1 0

1

2

3

kΔx

2 ⎛ ωΔt ⎞ 2 ⎛ k x Δx ⎞ 2 ⎛ k y Δy ⎞ ⎜ sin 2 ⎟ ⎜ sin 2 ⎟ ⎜ sin 2 ⎟ ⎟ ⎜ ⎟ = ⎜ ⎟ + ⎜ cΔt Δx Δy ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

€

17

Pushing Particles FSC

   E( x k ) = ∑ E ij S(x k − X ij )

•  To update p, EM fields have to first be interpolated from the grid to the particle locations

i, j

ΔA

Δy

–  Linear (area) weighting

Δx €

•  The same S should be used in charge deposition to ensure momentum conservation

 ΔA  S( x k − X ij ) = ΔxΔy   ∑ S(x k − X ij ) = 1

(i, j) €

€

i, j

€

€  ρ ij = ∑ qk S(x k − X ij ) k

 dP   = ∑ qk E(x k ) = ∑ qk ∑ E ij S(x k − X ij ) dt k k i, j

–  dP/dt=0 for 1 particle, independent of S.

= ∑ E ij ρ ij i, j

1

€

E1 = −

2

ρ2 ρ , E 2 = 1 ⇒ E1ρ1 + E 2 ρ 2 = 0 2 2 18

€

Pushing Particles (Boris Pusher) FSC

p n +1/ 2 − p n−1/ 2 q n 1 p n +1/ 2 + p n−1/ 2 = (E + × Bn ) n Δt m c 2γ B n +1/ 2 + B n−1/ 2 (B = ) 2

•  Differencing Relativistic Newton’s Eq.

n

•  Separating fE & fB

Δt q n E 2 m Δt q n p n +1/ 2 = p + + E 2 m p− = p n−1/ 2 +

€

•  The magnetic force causes a pure rotation €

p + − p− q p+ + p_ = × Bn n Δt mc 2γ [( p + ) 2 − ( p− ) 2 = 0]

€

19

€

v×B Rotation FSC

Eq. for v×B rotation p + − p− q p+ + p_ n = × B Δt mc 2γ n

p⊥+

p⊥+ − p−⊥

θ € €

− ⊥

tan

+ ⊥

p +p

€ p−⊥

θ qB =− Δt 2 2mcγ

€ €

In practice, this is used

€ p+

−

−

p'= p + p × t, p + = p− + p'×s,

θ (t = qBΔt /2γmc = −tan ) 2 θ θ (s = 2t /(1+ t 2 ) = −2sin cos ) 2 2

p'×s

θ p' €

€ p−

€

p− × t

€ €

€

20

Current Deposition & Charge Conservation FSC

x

• 

Updating positions

• 

Current deposition needs both V & x defined at different times

j

–  ∂ρ/∂t+∇⋅j=0 may not be satisfied

n +1/ 2 ij

€

= ∑q v

• 

Charge conservation is critical to guarantee solutions satisfying all Maxwell’s eqs. Two ways to achieve charge conservation

–  Adjusting longitudinal E –  Using charge-conserving current deposition schemes –  Spectral method

€

p n +1/ 2 = x + n +1/ 2 Δt γ

n +1/ 2 k k

k

or

• 

n +1

n

  x kn + x kn +1  S( − X ij ) 2

n   n +1  − X ij ) n +1/ 2 S( x k − X ij ) + S( x k

j ijn +1/ 2 = ∑ qk v k k

€

2

∂t B = −∇ × E ⇒ ∂t ∇ ⋅ B = 0 ∂t E = ∇ × B − J ⇒ ∂ t ∇ ⋅ E = −∇ ⋅ J ⎯ ⎯? →∂ t ρ

21

Charge-Conserving Current Deposit Scheme FSC

For unit grid & unit charge (0.5 − y) + (0.5 − y − Δy) 1 = Δx(0.5 − y − Δy) 2 2 1 J x 2 = Δx(0.5 + y + Δy) 2 1 J y1 = Δy(0.5 − x − Δx) 2 1 J y 2 = Δy(0.5 + x + Δy) 2 J x1 = Δx

€

For more complicated crossing, the current can be calculated in stages [see Villasenor & Buneman, Computer Phys. Comm. 69, 306 (1992).] 22

Adjusting E to Satisfy ∇⋅E=ρ FSC

•  E’ obtained from a non-CC j does not satisfy ∇⋅E’=ρ •  Find δφ so that ∇2 δφ= ∇⋅E’-ρ •  The corrected field is E=E’- ∇δφ

j+1 (Ez,, Bz, Jz)

j+1/2 (Ey,, By, Jy)

j

(ρ,δφ)

i

(Ex,, Bx, Jx)

i+1/2

i+1 23

Boundary conditions are important to model reality FSC

•  For fields –  Periodic –  Conducting –  Open, absorbing,…

•  For particles

z z z z E −1/ 2, j = E Nx−1/ 2, j , E Nx +1/ 2, j = E1/ 2, j ,...

E € €

y 0, j

z z E −1/ 2, j + E1/ 2, j = 0, =0 2

j+1 (Ez,, Bz, Jz)

j+1/2 (E , B , J ) (E , B , J ) j i i+1/2 i+1 y,

y

y

x,

x

x

–  Periodic –  Reflecting –  Thermal bath

24

Grid Instability FSC

• 

Since particle positions are continuous, δn can have high-k modes –  Dominated by kn~ωp/vt

• 

Due to aliasing, δρ can have modes with k=kn-p(2π/∆x)

• 

δρ can resonantly interact with E of the same k, causing instability

• 

In practice, if ∆ωp

overdense ω050 keV electron distribution in px-py space

Px

Px

Most hot e- are generated in the nonlinear stage FSC

10000

1000

100

2ps 4ps

10

9.9ps

1

The existence of >100 keV e- is consistent with hard X-ray diagnostic

The appearance of hot e- (>100 keV) is correlated to new modes that are not seen in the linear stage

Px

FSC

X

Plane-wave, 2D PIC simulations gave 10-100X more hot electrons than HXR measurements FSC Max I14

Te(keV)

Ti(kev)

Lµm

Mi/me

Run time

Forward Hot e-(>50kev)

Abs

η

4

3

1.5

150

3410

5ps

~0 (100 ptc per cell)

~0

0.8

6

3

1.5

150

3410

10ps

17% (100 ptc per cell)

42%

1.2

B. Yaakobi et al., Phys. Plasmas 16, 102703 (2009): hot e-